6 problems

Let $F$ be an irreducible holomorphic symplectic fourfold deformation equivalent to $S^{[2]}$ for a K3 surface $S$, and let the reflection group act on the positive cone of divisor…

Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and write a divisor class as … with the convention $b_i=b_{g-i}$. Here $\lambda$ is the Hodge clas…

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, let $NE_1(\overline{\mathcal{M}}_{g,n})$ denote its Mori cone of curves, and let a…

Ample-cone conjecture. If $C$ is a very general curve of genus $g\ge 4$, then

Let $d,m>0$, and let … be the divisor class on the blow-up $\pi:V_N\to {\mathbf C}P^2$ at $N\geq 9$ generic points. Nagata's conjecture. The divisor $D$ is ample if and only if…

For a projective variety $X$, let … be its cone of cylindrical ample divisors. The cylindrical ample cone conjecture. If $X$ is a Fano variety, then … The problem of determining th…